# Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just positive integers. The function is 1 if the variables are equal, and 0 otherwise:

${\displaystyle \delta _{ij}={\begin{cases}0&{\text{if }}i\neq j,\\1&{\text{if }}i=j.\end{cases}}}$

where the Kronecker delta δij is a piecewise function of variables i and j. For example, δ1 2 = 0, whereas δ3 3 = 1.

The Kronecker delta appears naturally in many areas of mathematics, physics and engineering, as a means of compactly expressing its definition above.

In linear algebra, the n × n identity matrix I has entries equal to the Kronecker delta:

${\displaystyle I_{ij}=\delta _{ij}}$

where i and j take the values 1, 2, ..., n, and the inner product of vectors can be written as

${\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i,j=1}^{n}a_{i}\delta _{ij}b_{j}.}$

The restriction to positive integers is common, but there is no reason it cannot have negative integers as well as positive, or any discrete rational numbers. If i and j above take rational values, then for example

{\displaystyle {\begin{aligned}\delta _{(-1)(-3)}&=0&\qquad \delta _{(-2)(-2)}&=1\\\delta _{\left({\frac {1}{2}}\right)\left(-{\frac {3}{2}}\right)}&=0&\qquad \delta _{\left({\frac {5}{3}}\right)\left({\frac {5}{3}}\right)}&=1.\end{aligned}}}

This latter case is for convenience.

## Properties

The following equations are satisfied:

{\displaystyle {\begin{aligned}\sum _{j}\delta _{ij}a_{j}&=a_{i},\\\sum _{i}a_{i}\delta _{ij}&=a_{j},\\\sum _{k}\delta _{ik}\delta _{kj}&=\delta _{ij}.\end{aligned}}}

Therefore, the matrix δ can be considered as an identity matrix.

Another useful representation is the following form:

${\displaystyle \delta _{nm}={\frac {1}{N}}\sum _{k=1}^{N}e^{2\pi i{\frac {k}{N}}(n-m)}}$

This can be derived using the formula for the finite geometric series.

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